DATA 606 - Lab 4 - Distributions of Random Variables

Distributions of Random Variables

https://htmlpreview.github.io/?https://github.com/jbryer/DATA606/blob/master/inst/labs/Lab4/Lab4_normal_distribution.html

library(tidyverse)
library(openintro)
library(DATA606)

## 
## Welcome to CUNY DATA606 Statistics and Probability for Data Analytics 
## This package is designed to support this course. The text book used 
## is OpenIntro Statistics, 4th Edition. You can read this by typing 
## vignette('os4') or visit www.OpenIntro.org. 
##  
## The getLabs() function will return a list of the labs available. 
##  
## The demo(package='DATA606') will list the demos that are available.

library(psych)

Exercise 1

Dairy Queen has more of a normal distribution of calories from fat when compared to McDonalds. Both are unimodal and right skewed, though McDonalds has the more pronounced right skew due to a large early cluster of values despite having a max value that is double Dairy Queen’s.

fastfood

## # A tibble: 515 × 17
##    restaurant item      calories cal_fat total_fat sat_fat trans_fat cholesterol
##    <chr>      <chr>        <dbl>   <dbl>     <dbl>   <dbl>     <dbl>       <dbl>
##  1 Mcdonalds  Artisan …      380      60         7       2       0            95
##  2 Mcdonalds  Single B…      840     410        45      17       1.5         130
##  3 Mcdonalds  Double B…     1130     600        67      27       3           220
##  4 Mcdonalds  Grilled …      750     280        31      10       0.5         155
##  5 Mcdonalds  Crispy B…      920     410        45      12       0.5         120
##  6 Mcdonalds  Big Mac        540     250        28      10       1            80
##  7 Mcdonalds  Cheesebu…      300     100        12       5       0.5          40
##  8 Mcdonalds  Classic …      510     210        24       4       0            65
##  9 Mcdonalds  Double C…      430     190        21      11       1            85
## 10 Mcdonalds  Double Q…      770     400        45      21       2.5         175
## # … with 505 more rows, and 9 more variables: sodium <dbl>, total_carb <dbl>,
## #   fiber <dbl>, sugar <dbl>, protein <dbl>, vit_a <dbl>, vit_c <dbl>,
## #   calcium <dbl>, salad <chr>

summary(fastfood)

##   restaurant            item              calories         cal_fat      
##  Length:515         Length:515         Min.   :  20.0   Min.   :   0.0  
##  Class :character   Class :character   1st Qu.: 330.0   1st Qu.: 120.0  
##  Mode  :character   Mode  :character   Median : 490.0   Median : 210.0  
##                                        Mean   : 530.9   Mean   : 238.8  
##                                        3rd Qu.: 690.0   3rd Qu.: 310.0  
##                                        Max.   :2430.0   Max.   :1270.0  
##                                                                         
##    total_fat         sat_fat         trans_fat      cholesterol    
##  Min.   :  0.00   Min.   : 0.000   Min.   :0.000   Min.   :  0.00  
##  1st Qu.: 14.00   1st Qu.: 4.000   1st Qu.:0.000   1st Qu.: 35.00  
##  Median : 23.00   Median : 7.000   Median :0.000   Median : 60.00  
##  Mean   : 26.59   Mean   : 8.153   Mean   :0.465   Mean   : 72.46  
##  3rd Qu.: 35.00   3rd Qu.:11.000   3rd Qu.:1.000   3rd Qu.: 95.00  
##  Max.   :141.00   Max.   :47.000   Max.   :8.000   Max.   :805.00  
##                                                                    
##      sodium       total_carb         fiber            sugar       
##  Min.   :  15   Min.   :  0.00   Min.   : 0.000   Min.   : 0.000  
##  1st Qu.: 800   1st Qu.: 28.50   1st Qu.: 2.000   1st Qu.: 3.000  
##  Median :1110   Median : 44.00   Median : 3.000   Median : 6.000  
##  Mean   :1247   Mean   : 45.66   Mean   : 4.137   Mean   : 7.262  
##  3rd Qu.:1550   3rd Qu.: 57.00   3rd Qu.: 5.000   3rd Qu.: 9.000  
##  Max.   :6080   Max.   :156.00   Max.   :17.000   Max.   :87.000  
##                                  NA's   :12                       
##     protein           vit_a            vit_c           calcium      
##  Min.   :  1.00   Min.   :  0.00   Min.   :  0.00   Min.   :  0.00  
##  1st Qu.: 16.00   1st Qu.:  4.00   1st Qu.:  4.00   1st Qu.:  8.00  
##  Median : 24.50   Median : 10.00   Median : 10.00   Median : 20.00  
##  Mean   : 27.89   Mean   : 18.86   Mean   : 20.17   Mean   : 24.85  
##  3rd Qu.: 36.00   3rd Qu.: 20.00   3rd Qu.: 30.00   3rd Qu.: 30.00  
##  Max.   :186.00   Max.   :180.00   Max.   :400.00   Max.   :290.00  
##  NA's   :1        NA's   :214      NA's   :210      NA's   :210     
##     salad          
##  Length:515        
##  Class :character  
##  Mode  :character  
##                    
##                    
##                    
## 

mcdonalds <- fastfood %>%
  filter(restaurant == "Mcdonalds")
dairy_queen <- fastfood %>%
  filter(restaurant == "Dairy Queen")

describe(mcdonalds$cal_fat)

##    vars  n   mean    sd median trimmed    mad min  max range skew kurtosis
## X1    1 57 285.61 220.9    240  248.94 118.61  50 1270  1220 2.27     6.34
##       se
## X1 29.26

hist(mcdonalds$cal_fat)

describe(dairy_queen$cal_fat)

##    vars  n   mean     sd median trimmed    mad min max range skew kurtosis
## X1    1 42 260.48 156.49    220  245.88 126.02   0 670   670 0.91     0.36
##       se
## X1 24.15

hist(dairy_queen$cal_fat)

Exercise 2

The distribution is very nearly a normal distribution except that it’s not exactly symmetrical. There are more values in the first quadrant than in the last quadrant when they should be representing equal populations and negligible populations at that.

dqmean <- mean(dairy_queen$cal_fat)
dqsd <- sd(dairy_queen$cal_fat)

ggplot(data = dairy_queen, aes(x = cal_fat)) +
  geom_blank() +
  geom_histogram(aes(y = ..density..)) +
  stat_function(fun = dnorm, args = c(mean = dqmean, sd = dqsd), col = "tomato")

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

Exercise 3

The simulations are similar and seem within a reasonable margin of error, but they seem to account for lower values between the -2 to -1 quantile than the real data does which shows more of a left skew in that most of it sighter values fall on the trend line.

set.seed(37)
sim_norm <- rnorm(n = nrow(dairy_queen), mean = dqmean, sd = dqsd)

ggplot(data = NULL, aes(sample = sim_norm)) +
  geom_line(stat = "qq")

qqnormsim(sim_norm)

hist(sim_norm)

Exercise 4

Most points in the real data set and simulated sets fall very consistently on their respective trend lines, which is supported by a histogram curve that very nearly matches the standard bell curve that tapers off on either end.

qqnormsim(dairy_queen$cal_fat)

hist(dairy_queen$cal_fat)

Exercise 5

The McDonalds probability plots show emphases towards values on the lower end with values on the higher end often being excluded from the trend line. Plotting a histogram, a right skew is very evident.

qqnormsim(mcdonalds$calories)

hist(mcdonalds$cal_fat)

Exercise 6

Dairy Queen is the best candidate for analyzing sugar and calcium since it’s primarily a dairy dessert destination. I expect that the probability of buying an item that will exceed half od one’s recommended sugar intake per day will be very high (12 grams)

sugarDailyServing <- 12

sugarMean <- mean(dairy_queen$sugar)
sugarSd <- sd(dairy_queen$sugar)

pnorm(sugarDailyServing, sugarMean, sugarSd)

## [1] 0.8692298

dairy_queen %>%
  filter(sugar < sugarDailyServing) %>%
  summarise(n() / nrow(dairy_queen))

## # A tibble: 1 × 1
##   `n()/nrow(dairy_queen)`
##                     <dbl>
## 1                   0.905

1 - pnorm(sugarDailyServing, sugarMean, sugarSd)

## [1] 0.1307702

dairy_queen %>%
  filter(sugar > sugarDailyServing) %>%
  summarise(n() / nrow(dairy_queen))

## # A tibble: 1 × 1
##   `n()/nrow(dairy_queen)`
##                     <dbl>
## 1                  0.0952

qqnormsim(dairy_queen$sugar)

hist(dairy_queen$sugar)

Exercise 7

Burger King and Arby’s have distributions that are the closest to the standard distribution model.

restaurantNames <- fastfood$restaurant %>%
  unique
restaurantNames

## [1] "Mcdonalds"   "Chick Fil-A" "Sonic"       "Arbys"       "Burger King"
## [6] "Dairy Queen" "Subway"      "Taco Bell"

for (x in 1:length(restaurantNames)) {
  restaurantName <- restaurantNames[x]
  restaurantSodium <- (fastfood %>%
    filter(restaurant == restaurantName))$sodium
  
  print(paste(restaurantName, "↓", sep = " "))
  qqnormsim(restaurantSodium)
  hist(restaurantSodium)
}

## [1] "Mcdonalds ↓"

## [1] "Chick Fil-A ↓"

## [1] "Sonic ↓"

## [1] "Arbys ↓"

## [1] "Burger King ↓"

## [1] "Dairy Queen ↓"

## [1] "Subway ↓"

## [1] "Taco Bell ↓"

Exercise 8

The steps indicate larger bins as opposed to smaller granular bins with many unique values that might better smooth the curve. That might indicate that salt is added or marketed in more controlled and consistent increments.

  summary(fastfood$sodium)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##      15     800    1110    1247    1550    6080

Exercise 9

Judging by the normal probability plot, the data set is right skewed because the trend line prioritizes lower values towards 0, which would account for values along the left side on the histogram. The resulting histogram confirms the skew in that in its overall range of 0-140, most values fall between a very early range of 20-50

qqnormsim(dairy_queen$total_carb)

hist(dairy_queen$total_carb)